The author, Darshak Sanghavi argues that no, TV and Video Games do not make kids fat. According to Sanghavi:

“If video games aren’t the problem, then what about television? We’ve know for a long time that attempts to reduce television-watching among children have a limited effect on their body weight. For a 1999 paper in the Journal of the American Medical Association, researchers gave a group of third- and fourth-graders in California regular lessons on the dangers of excessive television. Their parents were asked to enforce time budgets (using a device to limit total screen time) and participate in television turnoffs lasting 10 days, among other projects. This very involved, two-month intervention halved television watching among participants. Eight months later, researchers measured the children’s heights and weights, and compared them to those taken from children at a school without a similar program. The drastic reduction in television-watching made for only a very modest difference: Weight gains in the experimental group were reduced by an average of only one pound.”

But 8 to 10-year-olds are still growing. Just purely from growth they would be expected to gain weight. That is why the researchers, who did this study, did not use weight as their main metric. From the link that Sanghavi provided, this is how the researchers described their results:

Journal of the American Medical Association 1999

“Results: Compared with controls, children in the intervention group had statistically significant relative decreases in body mass index… , triceps skinfold thickness… , waist circumference… , and waist-to-hip ratio… . Relative to controls, intervention group changes were accompanied by statistically significant decreases in children’s reported television viewing and meals eaten in front of the television. There were no statistically significant differences between groups for changes in high-fat food intake, moderate-to-vigorous physical activity, and cardiorespiratory fitness.”

Conclusions: Reducing television, videotape, and video game use may be a promising, population-based approach to prevent childhood obesity.”